The Divisor function The divisor function counts the number of divisors of an integer Dirichlet divisor problem Determine the asymptotic behaviour as of the sum This is a count of lattice points under the hyperbola ID: 269866
Download Presentation The PPT/PDF document "Arithmetic Statistics in Function Fields" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
Slide1
Arithmetic Statistics in Function FieldsSlide2
The Divisor function
The
divisor function counts the number of divisors of an integer . Dirichlet divisor problem:Determine the asymptotic behaviour as of the sum This is a count of lattice points under the hyperbola
Slide3
Counting method
Make use of the symmetry of the region about the line
count twice the number of lattice points under the hyperbola and under the line . * On the diagonal line there are lattice points.* On horizontal lines we have lattice points. Slide4
Therefore
Use
and the Euler summation formula
Slide5
Dirichlet
‘s divisor problem:
Dirichlet: Voronoi (1903): Huxley (2003):Problem (Divisor function in short intervals): The limiting distribution of when .
The trivial range:
For
and
, as
Slide6
The Divisor function in short intervals
Define
=Ivic (2009): For with a certain cubic polynomial. Slide7
The generalized Divisor function
The
k-th divisor function : The classical divisor function being Example: for a prime number , =
Generalization of
Dirichlet
‘s divisor problem:
,
where
is a certain polynomial of degree
Slide8
The generalized Divisor function in short intervals
Lester
(2015):, then (assuming ) Conjecture J.P.Keating, B.Rodgers, ER-G and Z.Rudnick (2015) ,
,
then
Where
is a piecewise polynomial function in
,
of degree
Slide9
Function fields
a finite field ( is a power of an odd prime) the ring of polynomials with coefficients in . the set of polynomials of degree be the subset of monic polynomials.The norm of
is
defined by
.
The k-
th
divisor function
Compare with number fields:
Slide10
Divisors in function fields
The analogue of
Dirichlet divisor problem:Over function fields- an easy computation:The k-th power of the zeta function is the generating function of
Expand + compare the coefficient of
.
Slide11
Short intervals in
For and an interval around of length is Note that .The sum over “short interval” The mean value is
Define
Our goal
:
to
study
the variance
of
(in the limit of a large field size)
Slide12
Short intervals as arithmetic progressions
There is a bijection between Intervals and arithmetic progressions:
When , deg This covers all intervals since Slide13
Variance in short intervals
Theorem (
J.P.Keating, B.Rodgers, E.R-G and Z.Rudnick 2015) Let and , then as Where
and
are the secular
coefficients:
Corollary:
If
and
, then as
Slide14
Comparison between number field and function field results (short intervals)
Number field
Function field If then
Number field
Function fieldSlide15
Ingredients of the proof
Orthogonality relation for
Dirichlet characters mod Q: Even characters for all
Riemann Hypothesis (proved by Weil) :
Spectral interpretation (for primitive even characters):
,
Slide16
The main ingredient
Theorem
(Nick Katz 2013)The unitarized Frobenii when is an even primitive character mod become equidistributed in PU(m-1) as
Slide17
Sketch of the proof
evaluated in terms of evaluate in terms of the associated Dirichlet L- functionsWrite the L-functions in terms of unitary matrices The variance
Slide18
Sketch of the proof
Apply Katz
equidistribution result
Slide19
Matrix Integral
What do we know about
?For , Functional equation
Slide20
Theorem (
J.P.Keating
, B.Rodgers, ER-G and Z.Rudnick) is equal to the count of lattice points satisfying each of the relations for all
, where
is the collection of
matrices whose entries satisfy the following system of equalities,
Slide21
Let
. Then for
With
Here
is the Barnes G-function, so that for positive integers k,
is a piecewise polynomial which changes when ever
reaches an integer.
Slide22
Thank you!